Right Coset Equals Subgroup iff Element in Subgroup

Theorem

Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Let $x \in G$.

Let $H x$ denote the right coset of $H$ by $x$.

Then:

$H x = H \iff x \in H$

Proof

\(\ds H x\)

\(=\)

\(\ds H\)

\(\ds \leadstoandfrom \ \ \)

\(\ds H x\)

\(=\)

\(\ds H e\)

Right Coset by Identity: $H = H e$

\(\ds \leadstoandfrom \ \ \)

\(\ds x e^{-1}\)

\(\in\)

\(\ds H\)

Right Cosets are Equal iff Product with Inverse in Subgroup

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds H\)

Group Properties

$\blacksquare$

Also see

• Left Coset Equals Subgroup iff Element in Subgroup

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Example $112$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(3R)}$ Another approach to cosets
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $7 \ \text{(iii)}$